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 Title: Annular breadth of hinges & hinge exit paths of annuli Author(s): Tichenor, Scott R. Director of Research: Alexander, Stephanie Doctoral Committee Chair(s): Reznick, Bruce Doctoral Committee Member(s): Wetzel, John E; Bishop, Richard Department / Program: Mathematics Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): exit path escape path width annulus polygonal arc breadth Wetzel broadworm Voronoi Rivlin Abstract: Given a compact set $\textsf{S}\subset\mathds{R}^2$, we define the annular width function for $\textsf{S}$, denoted $w(E)$, as the width of the annulus of support of $\textsf{S}$ centered at $E\in\overline{\mathds{R}^2}$, where $\overline{\mathds{R}^2}$ is an extension of the real plane $\mathds{R}^2$. The annular breadth of $\textsf{S}$ is defined as the absolute minimum of $w(E)$. We find the $2$-segment polygonal arc with the greatest annular breadth. For a given set $\textsf{S}\subset\mathds{R}^2$, an exit path of $\textsf{S}$ is a curve that cannot be covered by the interior of $\textsf{S}$. Given an annulus, we find its shortest $1$- or $2$-segment polygonal arc exit path(s). Bezdek and Connelly provided a lengthy and technically demanding proof that \emph{All orbiforms of width} $1$ \emph{are translation covers of the set of closed planar curves of length} $2$ \emph{or less}. We provide a short and simple proof that \emph{All orbiforms of width} $1$ \emph{are covers of the set of all planar curves of length} $1$ \emph{or less}. We also provide a proof that \emph{The Reuleaux triangle of width} $1$ \emph{is a cover of the set of all closed curves of length} $2$ using a recent of Wichiramala. Issue Date: 2019-07-03 Type: Text URI: http://hdl.handle.net/2142/105618 Rights Information: Copyright 2019 by Scott R. Tichenor. All rights reserved. Date Available in IDEALS: 2019-11-26 Date Deposited: 2019-08
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